x = seq(-3, 3, by = 0.1)
plot(x, pnorm(x), type = "l")
x=-3:0.1:3;
plot(x,normcdf(x))
title("CDF of Normal Distribution")
set.seed(12) # set seed
S = 10
runif(S)
rnorm(S)
rt(S,4)
rng default; % set seed
S=10;
rand(S,1)
randn(S,1)
trnd(4,S,1)
yield = c(5.00, 5.69, 6.09, 6.38, 6.61,
6.79, 6.94, 7.07, 7.19, 7.30) # yield curve
T = length(yield) # number of time periods
r = 0.07 # initial yield rate
Par = 10 # par value
coupon = r * Par # coupon payments
cc = rep(coupon, T) # vector of cash flows
cc[T] = cc[T] + Par # add par to cash flows
P = sum(cc/((1+yield/100)^(1:T))) # calculate price
print(P)
yield = [5.00 5.69 6.09 6.38 6.61...
6.79 6.94 7.07 7.19 7.30]; % yield curve
T = length(yield);
r=0.07; % initial yield rate
Par=10; % par value
coupon=r*Par; % coupon payments
cc=zeros(1,T)+coupon; % vector of cash flows
cc(T)=cc(T)+Par; % add par to cash flows
P=sum(cc./((1+yield./100).^(1:T))) % calculate price
set.seed(12) # set seed
sigma = 1.5 # daily yield volatiltiy
S = 8 # number of simulations
r = rnorm(S, 0, sigma) # generate random numbers
ysim = matrix(nrow=length(yield),ncol=S)
for (i in 1:S) ysim[,i]=yield+r[i]
matplot(ysim,type='l')
randn('state',123); % set the seed
sigma = 1.5; % daily yield volatility
S = 8; % number of simulations
r = randn(1,S)*sigma; % generate random numbers
ysim=nan(T,S);
for i=1:S
ysim(:,i)=yield+r(i);
end
ysim=repmat(yield',1,S)+repmat(r,T,1);
plot(ysim)
title("Simulated yield curves")
SP = rep(NA, length = S)
for (i in 1:S){ # S simulations
SP[i] = sum(cc/((1+ysim[,i]/100)^(1:T)))
}
SP = SP-(mean(SP) - P) # correct for mean
par(mfrow=c(1,2), pty="s")
barplot(SP)
hist(SP,probability=TRUE)
x=seq(6,16,length=100)
lines(x, dnorm(x, mean = mean(SP), sd = sd(SP)))
S = 50000
r = rnorm(S, 0, sigma) # generate random numbers
ysim = matrix(nrow=length(yield),ncol=S)
for (i in 1:S) ysim[,i]=yield+r[i]
SP = rep(NA, length = S)
for (i in 1:S){ # S simulations
SP[i] = sum(cc/((1+ysim[,i]/100)^(1:T)))
}
SP = SP-(mean(SP) - P) # correct for mean
par(mfrow=c(1,2), pty="s")
barplot(SP)
hist(SP,probability=TRUE)
x=seq(6,16,length=100)
lines(x, dnorm(x, mean = mean(SP), sd = sd(SP)))
SP = nan(S,1);
for s = 1:S % S simulations
SP(s) = sum(cc./(1+ysim(:,s)'./100).^((1:T)));
end
SP = SP-(mean(SP) - P); % correct for mean
bar(SP)
S = 50000;
rng("default")
r = randn(S,1) * sigma;
ysim = nan(T,S);
for i = 1:S
ysim(:,i) = yield' + r(i);
end
SP = nan(S,1);
for i = 1:S
SP(i) = sum(cc./(1+ysim(:,i)'./100).^((1:T)));
end
SP = SP - (mean(SP)-P);
histfit(SP)
title("Histogram of simulated bond prices with fitted normal")
P0 = 50 # initial spot price
sigma = 0.2 # annual volatility
r = 0.05 # annual interest
TT = 0.5 # time to expiration
X = 40 # strike price
f = bs(X = X, P = P0, r = r, sigma = sigma, T = TT) # analytical call price
## This calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
print(f)
P0 = 50; % initial spot price
sigma = 0.2; % annual volatility
r = 0.05; % annual interest
T = 0.5; % time to expiration
X = 40; % strike price
f = bs(X,P0,r,sigma,T) % analytical call price
%% This calculation uses the Black-Scholes pricing function (Listing 6.1/6.2)
set.seed(12) # set seed
S = 1e6 # number of simulations
F = P0*exp(r*TT) # futures price
ysim = rnorm(S,-0.5*sigma^2*TT,sigma*sqrt(TT)) # sim returns, lognorm corrected
F = F*exp(ysim) # sim futures price
SP = F-X # payoff
SP[SP<0] = 0 # set negative outcomes to zero
fsim = SP*exp(-r*TT) # discount
call_sim = mean(fsim) # simulated price
print(call_sim)
randn('state',0); % set seed
S = 1e6; % number of simulations
ysim = randn(S,1)*sigma*sqrt(T)-0.5*T*sigma^2; % sim returns, lognorm corrected
F = P0*exp(r*T)*exp(ysim); % sim future prices
SP = F-X; % payoff
SP(find(SP < 0)) = 0; % set negative outcomes to zero
fsim = SP * exp(-r*T) ; % discount
mean(fsim) % simulated price
par(mfrow=c(1,2), pty="s")
hist(F, probability=TRUE, ylim=c(0,0.06))
x = seq(min(F), max(F), length=100)
lines(x, dnorm(x, mean = mean(F), sd = sd(SP)))
hist(fsim, nclass=100, probability=TRUE)
subplot(1,2,1)
histfit(F);
title("Simulated prices");
xline(X, 'LineWidth', 1, 'label', 'Strike');
subplot(1,2,2)
hist(fsim,100);
title("Option price density");
xline(mean(fsim), 'LineWidth', 1, 'label', 'Call');
set.seed(1) # set seed
S = 1e7 # number of simulations
s2 = 0.01^2 # daily variance
p = 0.01 # probability
r = 0.05 # annual riskfree rate
P = 100 # price today
ysim = rnorm(S,r/365-0.5*s2,sqrt(s2)) # sim returns
Psim = P*exp(ysim) # sim future prices
q = sort(Psim-P) # simulated P/L
VaR1 = -q[p*S]
print(VaR1)
randn('state',0); % set seed
S = 1e7; % number of simulations
s2 = 0.01^2; % daily variance
p = 0.01; % probability
r = 0.05; % annual riskfree rate
P = 100; % price today
ysim = randn(S,1)*sqrt(s2)+r/365-0.5*s2; % sim returns
Psim = P*exp(ysim); % sim future prices
q = sort(Psim-P); % simulated P/L
VaR1 = -q(S*p)
TT = 0.25 # time to expiration
X = 100 # strike price
sigma = sqrt(s2*250) # annual volatility
f = bs(X, P, r, sigma, TT) # analytical call price
fsim = bs(X,Psim,r,sigma,TT-(1/365)) # sim option prices
q = sort(fsim$Call-f$Call) # simulated P/L
VaR2 = -q[p*S]
print(VaR2)
T = 0.25; % time to expiration
X = 100; % strike price
sigma = sqrt(s2*250); % annual volatility
f = bs(X,P,r,sigma,T); % analytical call price
fsim=bs(X,Psim,r,sigma,T-(1/365)); % sim option prices
q = sort(fsim.Call-f.Call); % simulated P/L
VaR2 = -q(p*S)
X1 = 100
X2 = 110
f1 = bs(X1, P, r, sigma, TT)
f2 = bs(X2, P, r, sigma, TT)
f2sim = bs(X2, Psim, r, sigma, TT-(1/365))
f1sim = bs(X1, Psim, r, sigma, TT-(1/365))
q = sort(f1sim$Call + f2sim$Put + Psim-f1$Call - f2$Put-P);
VaR3 = -q[p*S]
print(VaR3)
X1 = 100;
X2 = 110;
f1 = bs(X1,P,r,sigma,T);
f2 = bs(X2,P,r,sigma,T);
f1sim=bs(X1,Psim,r,sigma,T-(1/365));
f2sim=bs(X2,Psim,r,sigma,T-(1/365));
q = sort(f1sim.Call+f2sim.Put+Psim-f1.Call-f2.Put-P);
VaR3 = -q(p*S)
library (MASS)
set.seed(12) # set seed
mu = rep(r/365, 2) # return mean
Sigma = matrix(c(0.01, 0.0005, 0.0005, 0.02),ncol=2) # covariance matrix
y = mvrnorm(S,mu,Sigma) # simulated returns
randn('state',12) % set seed
mu = [r/365 r/365]'; % return mean
Sigma=[0.01 0.0005; 0.0005 0.02]; % covariance matrix
y = mvnrnd(mu,Sigma,S); % simulated returns
K=2
P = c(100, 50) # prices
x = rep(1, 2) # number of assets
Port = P %*% x # portfolio at t
Psim = matrix(t(matrix(P,K,S)),ncol=K)*exp(y) # simulated prices
PortSim = Psim %*% x # simulated portfolio value
q = sort(PortSim-Port[1,1]) # simulated P/L
VaR4 = -q[S*p]
print(VaR4)
K = 2;
P = [100 50]'; % prices
x = [1 1]'; % number of assets
Port = P'*x; % portfolio at t
Psim = repmat(P,1,S)' .*exp(y); % simulated prices
PortSim=Psim * x; % simulated portfolio value
q = sort(PortSim-Port); % simulated P/L
VaR4 = -q(S*p)
f = bs(X = P[2], P = P[2], r = r, sigma = sigma, T = TT)
fsim = bs(X = P[2], P = Psim[,2], r = r, sigma = sigma, T = TT-(1/365))
q = sort(fsim$Call + Psim[,1] - f$Call - P[1]);
VaR5 = -q[p*S]
print(VaR5)
f = bs(P(2),P(2),r,sigma,T);
fsim=bs(P(2),Psim(:,2),r,sigma,T-(1/365));
q = sort(fsim.Call+Psim(:,1)-f.Call-P(1));
VaR5 = -q(p*S)